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Chapter 10: Problem 50

Find the squares of \(2 e^{i k / 3}=1+\sqrt{3} i\) and \(4 e^{i \pi / 4}=\) \(2 \sqrt{2}+i 2 \sqrt{2}\) in both polar and rectangular coordinates.

Short Answer

Expert verified
The squares are 1) \(-2 + 2\sqrt{3}i\) in rectangular and \(4e^{i \frac{2\pi}{3}}\) in polar, 2) \(8i\) in rectangular and \(16e^{i \frac{\pi}{2}}\) in polar.

Step by step solution

01

Understand the Given Complex Numbers

We have two complex numbers given in both exponential and rectangular forms: For the first number: - Exponential form: \(2 e^{i \frac{k}{3}} \) - Rectangular form: \(1+\sqrt{3} i\) For the second number: - Exponential form: \(4 e^{i \frac{\pi}{4}} \) - Rectangular form: \(2\sqrt{2} + i 2\sqrt{2}\).
02

Verify the Conversion from Polar to Rectangular

Convert the exponential form to rectangular form to verify:1. **First Number**: - The given form is \(2 e^{i \frac{k}{3}} = 1 + \sqrt{3} i\). - The magnitude \(r = 2\), with angle \(\theta = \frac{k}{3}\). Solving \(\cos\theta = \frac{1}{2}\) and \(\sin\theta = \frac{\sqrt{3}}{2}\), gives \(\theta = \frac{\pi}{3}\). - So, \( k = \pi \). 2. **Second Number**: - The given form is \(4 e^{i \frac{\pi}{4}} = 2\sqrt{2} + i 2\sqrt{2}\). - The magnitude \(r = 4\), with angle \(\theta = \frac{\pi}{4}\). The given rectangular coordinates match.
03

Square the Complex Numbers in Rectangular Form

1. **First Number (Rectangular)**: - \((1 + \sqrt{3}i)^2 = 1^2 + 2(1)(\sqrt{3}i) + (\sqrt{3}i)^2 = 1 + 2\sqrt{3}i - 3 = -2 + 2\sqrt{3}i\).2. **Second Number (Rectangular)**: - \((2\sqrt{2} + i 2\sqrt{2})^2 = (2\sqrt{2})^2 + 2(2\sqrt{2})(i 2\sqrt{2}) + (i 2\sqrt{2})^2 = 8 + 8i - 8 = 8i\).
04

Square the Complex Numbers in Polar Form

1. **First Number (Polar)**: - \((2e^{i\frac{\pi}{3}})^2 = 2^2 e^{i \frac{2\pi}{3}} = 4 e^{i \frac{2\pi}{3}}\).2. **Second Number (Polar)**: - \((4 e^{i \frac{\pi}{4}})^2 = 4^2 e^{i \frac{\pi}{2}} = 16 e^{i \frac{\pi}{2}}\).
05

Convert the Squared Results Back to Rectangular Form

1. **First Squared (Polar to Rectangular)**: - \(4 e^{i \frac{2\pi}{3}} = 4(\cos\frac{2\pi}{3} + i \sin\frac{2\pi}{3}) = -2 + 2\sqrt{3}i\).2. **Second Squared (Polar to Rectangular)**: - \(16 e^{i \frac{\pi}{2}} = 16(0 + i 1) = 16i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
In mathematics, especially in trigonometry and complex numbers, polar coordinates represent a point in terms of a radius and an angle. This form is particularly useful in situations involving rotations and periodic functions.
Complex numbers can be represented in polar coordinates as \( r e^{i\theta} \), where \( r \) is the magnitude (or radius), and \( \theta \) is the angle measured from the positive x-axis.
  • The magnitude \( r \) is the distance from the origin to the point.
  • The angle \( \theta \) represents the direction of the line from the origin to the point.
Polar coordinates make it easier to perform operations such as multiplication and division of complex numbers. Here, multiplication corresponds to multiplying the magnitudes and adding the angles.
Rectangular Coordinates
Rectangular coordinates express a complex number in terms of its horizontal and vertical components on the complex plane. This is the more familiar way of expressing complex numbers: \( a + bi \), where:
  • \( a \) is the real part, corresponding to the x-coordinate.
  • \( b \) is the imaginary part, corresponding to the y-coordinate.
Using rectangular coordinates provides a straightforward way to visualize complex numbers as points on the plane, using the real axis (horizontal) and imaginary axis (vertical). While polar coordinates are useful for multiplication, rectangular coordinates are typically simpler for addition and subtraction.
Conversion Between Forms
To fully leverage both polar and rectangular coordinates, converting between these forms is a fundamental skill.
Understanding conversion helps in problem-solving as certain operations are easier in specific forms.
  • From rectangular \( a + bi \) to polar \( re^{i\theta} \):
    • Calculate \( r = \sqrt{a^2 + b^2} \) to find the magnitude.
    • Determine \( \theta = \arctan\left(\frac{b}{a}\right) \) to find the angle.
  • From polar \( re^{i\theta} \) to rectangular \( a + bi \):
    • Use \( a = r \cos\theta \) and \( b = r \sin\theta \).
With practice, this conversion becomes second nature, allowing fluid movement between different forms of complex numbers.
Squaring Complex Numbers
Squaring complex numbers is a common operation and can be performed in both rectangular and polar forms. Let's explore both.
  • Rectangular Form \((a + bi)^2 \):
    • Apply the formula: \( (a + bi)^2 = a^2 + 2abi + (bi)^2 \).
    • Simplify further using \( i^2 = -1 \): \( a^2 - b^2 + 2abi \).
  • Polar Form \((re^{i\theta})^2 \):
    • Simply square the magnitude: \( r^2 \).
    • Double the angle: \( 2\theta \), which gives \( r^2e^{2i\theta} \).
Understanding both methods ensures flexibility and efficiency, allowing you to choose the most convenient form depending on the problem context.

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